Question

Inside a spacecraft travelling at a speed of 0.8c with respect to the earth, the time
interval between two events on the earth is measured to be 30 hrs. What would the
time interval be if the spacecraft was travelling at a speed of 0.9c.

Answer 1

Here length contraction will take place when we travel with speed comparable to speed of light

$L=L_o\sqrt{1-\frac{v^2}{c^2}}$

$L=L_o\sqrt{1-\frac{(0.8c)^2}{c^2}}=L_o\times 0.6$

$t=L/v\\t=0.6L_o/0.8c\\L_o=\frac{0.8c\times30hrs}{0.6}=40c.hrs$

in the second case the speed is 0.9c then

$L=L_o\sqrt{1-\frac{v^2}{c^2}}$

$L=L_o\sqrt{1-\frac{0.81c^2}{c^2}}=0.43L_o$

$t=\frac{L}{v}=\frac{0.43L_o}{0.9c}=\frac{0.43\times40c.hrs}{0.9c}=19.11 hrs$